Fatima starts from point P. goes North for 3 km, and then East for 4 km to reach point Q. She then turns to face point P and goes 15 km in that direction. She then goes North for 6 km. How far is she from point P, and in which direction should she go to reach point P?

Question

Prepp Team · Accepted Answer

Step-by-Step Solution for Fatima's Journey

We can solve this problem using coordinate geometry by tracking Fatima's position relative to her starting point P.

1. Initial Movements and Position Q

Let the starting point P be the origin (0, 0) on a coordinate plane.
Fatima moves North 3 km. Her position is now (0, 3).
She then moves East 4 km. Her position becomes (0 + 4, 3) = (4, 3). This is point Q.

So, point Q is located at coordinates (4, 3).

2. Movement Towards Point P

Fatima is at Q(4, 3) and turns to face P(0, 0).
The vector representing the direction from Q to P is $P - Q = (0 - 4, 0 - 3) = (-4, -3)$.
The distance between P and Q is the magnitude of this vector: $\|QP\| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ km.
Fatima travels 15 km in the direction from Q towards P. The displacement vector for this movement is $15 \times \frac{P-Q}{\|QP\|} = 15 \times \frac{(-4, -3)}{5} = 3 \times (-4, -3) = (-12, -9)$.
Her new position after this movement is Q + displacement = (4, 3) + (-12, -9) = (4 - 12, 3 - 9) = (-8, -6).

3. Final Movement and Position

From the position (-8, -6), Fatima moves North 6 km.
Her final coordinates are (-8, -6 + 6) = (-8, 0).

4. Final Distance and Direction from P

Fatima's final position is (-8, 0). Her starting point P is (0, 0).
The distance from her final position to P is calculated using the distance formula: $\sqrt{(-8 - 0)^2 + (0 - 0)^2} = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8$ km.
Her final position (-8, 0) is 8 km directly West of P(0, 0).
To reach point P (0, 0) from her final position (-8, 0), she needs to move 8 km in the East direction.

Therefore, she is 8 km from point P, and she should go East to reach point P.